Surface Area/Transcript
Transcript Text reads: The Mysteries of Life with Tim and Moby. Nat and Moby stand behind a gift-wrapping booth in a department store. They wear uniforms and have measuring tape draped around their necks. The store manager waves at them, and they wave back. NAT: She knows we've got zero gift-wrapping experience, right? MOBY: Beep. Moby shrugs. NAT: Well, here goes nothing… Nat flips their sign on their booth from "closed" to "open." A customer approaches and places several boxes of perfume on the counter. A receipt with a letter is taped to one of the boxes. NAT: Wow, that's a lot of fragrance. Nat reads from the letter on the receipt. NAT: Dear Nat and Moby, How do you calculate surface area—and how is it useful? From, Peacock Leader. Great question Peacock Leader! The Surface area is the total space on the faces of a three-dimensional object. Like this box of Eau de Water. Nat holds one of the fragrance boxes. NAT: Knowing the surface area will tell us how much gift wrap we'll use to cover it. An animation shows the pre-cut wrapping paper at Nat and Moby's booth. The paper sizes are labeled as "200 square inches," "300 square inches," and "1000 square inches." NAT: And Boss Lady said we better not use any more than we need, or else! Nat draws her finger across her throat. Moby clutches his own throat in surprise. MOBY: Beep? NAT: Well, this box is a kind of prism: A three-dimensional shape that has two identical, parallel faces. An animation shows several types of prisms. The identical faces light up. NAT: These bases tell you the type of prism you're dealing with. Like, this one's an octagonal prism, because the bases are octagons. The animation shows an octagonal prism. The eight-sided bases light up. NAT: And this box is a rectangular prism, because the bases are rectangles. The animation shows the rectangular prism. MOBY: Beep? NAT: Uh, right, rectangular prisms are a little special. They're made up entirely of identical, parallel faces. So really, you could call any of them the base. Each pair of opposite faces lights up. There are three pairs of opposite faces. CUSTOMER: Ahem! An animation shows the impatient customer. He looks at his watch. NAT: Alrighty, let's get to wrapping! First, we'll want to measure the edges of our prism. Nat holds her tape measure to each edge of the prism. NAT: This box is four inches long… by two inches wide… by 10 inches tall. Labels appear on each edge of the prism. NAT: The simplest way to calculate surface area is to add up the areas of all the faces. We can do that by unfolding the object into a 2D shape called a net. The prism unfolds and lays flat. There are six rectangles: one for each face of the prism. The customer grumbles. NAT: Don't worry, we'll put it back together… eventually. Nat holds the flattened perfume box. Moby sniffs the perfume and then spritzes himself with it. NAT: Now we find each rectangle's area, multiplying length by width. The area calculations appear inside each rectangle in the net. Two rectangles show the equation "four times two equals eight square inches." Another two rectangles show the equation "two times 10 equals 20 square inches." The last two rectangles show the equation "four times 10 equals 40 square inches." NAT: Now, add up all the rectangles' areas—in whatever order: So, the total surface area is... 136 square inches. The animation shows the calculation "eight plus 20 plus 40 plus 40 plus 20 plus eight equals 136 square inches." MOBY: Beep? NAT: Yup, there are all kinds of shortcuts you can take, depending on the shape. Remember, this kind of prism is made entirely of identical pairs of rectangles. An animation shows the net of the rectangle. Two rectangles have an area of eight inches squared, two have an area of 20 inches squared, and two have an area of 40 inches squared. NAT: So, we could take the area of one of each pair; add them up; and multiply by two. The animation shows the calculation "eight plus 40 plus 20 times two equals 136." NAT: Boom: 136, just like we got the long way around! An animation shows the customer, who lets out a long sigh. NAT: Sorry, sir, my associate will get you all sorted out! Nat hands the perfume box to Moby. He grabs a piece of 200 inches squared paper and wraps it in a flurry. The wrapped gift looks terrible. The customer frowns and takes his boxes away. NAT: Ahh, our first satisfied customer! Another customer approaches Nat and Moby's booth. She places a large chocolate bar on the counter. MOBY: Beep! NAT: We can do it, Moby—this big hunk o' chocolate is a triangular prism. Let's try calculating its surface area without opening the box. We'll start again by taking measurements of all the edges. An animation shows the basic shape of a triangular prism. The edges of the triangular base measure three inches, four inches, and five inches. The height of the prism is 20 inches. NAT: Then we'll tackle finding the area of the triangular bases. The triangular bases of the prism light up. The three-inch and four-inch sides of the triangles meet at a right angle. NAT: Base times height, divided by two… six square inches. The base and height light up. The animation shows the calculation "three times four divided by two equals six square inches." NAT: Next, find the area of the other three faces: the rectangles. Just multiply length times width for each one, and add them all up. Each rectangular face lights up. One face is labeled "20 times three," another is labeled "20 times four," and another is labeled "20 times 5." Moby notices that all the calculations involve multiplying by 20. MOBY: Beep? NAT: Actually, yeah, we could calculate the non-base areas all at once. Since they all have the same length, they can unfold into one big rectangle. An animation shows the triangular prism. It transforms into its net. The net has three rectangles side-by-side and two triangles connected at each end. The animation outlines the large rectangle created by the three smaller rectangles of the same length. NAT: Whose width is the perimeter of the base. The widths of the individual rectangles are labeled as "three inches," "four inches," and "five inches." The labels disappear and are replaced with one that measures the width of all three rectangles together. This label reads "20 inches." NAT: So, the area of those three faces is 240. The animation shows the calculation "12 times 20 equals 240 square inches." NAT: Add that to the area of the two bases, and we get 252 square inches. The animation shows the sum that Nat describes. MOBY: Beep! An animation shows Moby at the wrapping booth. Moby wraps the bar of chocolate using a 300-square-inch piece of paper. This gift looks worse than the last one. The wrapping is even torn in some spots. Moby holds it up for Nat to see. NAT: Yeah… just stick some ribbon over the ripped parts. Nat attaches ribbon to the gift and hands it to the customer. The customer walks away. NAT: Happy holidays! Another customer approaches the gift-wrapping booth. She is carrying a round hatbox. NAT: Well, lookie here—those circular bases tell us that this is a cylinder. Moby opens the lid of the box. It has a gaudy Kentucky Derby hat inside. NAT: Holy Derby Day! Anyway, to find the surface area of this box, we'll need to measure the diameter and the height. An animation transforms the hatbox into a plain cylinder. The diameter is labeled as "20 inches" and the height is labeled as "six inches." NAT: Use the pi r squared formula to calculate the area of the circular base. The animation shows the circle area calculation. 10 inches is the value of r, because radius is half of the diameter. The resulting area equals 314 square inches. NAT: Then we double that, since there are two bases. The animation shows the calculation "314 times two equals 628 square inches." NAT: The cylinder’s other face is really just a rolled-up rectangle. The animation separates the cylinder into its parts. The net is made up of two circles and one rectangle. NAT: So, to find the area, multiply length by width. The animation reassembles the cylinder. The length of the rectangle connects to the edge of the circles perfectly. NAT: Which is actually the same as multiplying the base’s circumference by the cylinder’s height. The circle's circumference is labeled "pi times D." It changes to "pi times 20," and then "62.8 inches." The height is labeled "six inches." The animation shows the calculation "six times 62.8 equals 376.8 square inches." NAT: Finally, add those two numbers, for a total surface area of 1004.8 square inches. The animation shows "628 plus 376.8 equals 1004.8 square inches." MOBY: Beep. Moby points to the pre-cut wrapping paper. The largest size they have is 1000 square inches. NAT: Darn, just a little short! Moby removes the hat from its box and places it on the customer's head. He hands her the hatbox without bothering to wrap it. NAT: Hey, there you go—gift-wrapping hats is bad luck anyway. Thank you, next! The customer with the hat walks away. A child carrying an armful of dice approaches the booth. He dumps the dice onto the counter. Moby measures the length of one of the dice. MOBY: Beep! NAT: These are cubes. The sides of a cube are all the same length, so all six faces are identical, too. An animation shows a model cube. Each face lights up. NAT: The sides of this cube are all one inch, so each face is one square inch. The animation labels the side of the cube and the area of one of the faces. NAT: And there are six faces, for a total surface area of six square inches. The animation shows the calculation "one times six equals six." MOBY: Beep! Moby gives Nat a thumbs up and tries to leave the booth. Nat calls after him before he can walk away. NAT: Hold up, now, there're like 100 dice here! CHILD: 125. NAT: Wrapping each individually would take a lot of effort and paper. six square inches times 125 is… 750 square inches of paper! How about we arrange all of the dice together—into one big five-by-five-by-five cube. Nat quickly arranges the dice into a large cube. The cube has a length, width, and height of five dice. NAT: Fewer surfaces exposed means less surface area, which'll save us on paper. Each face of the cube is 25 square inches, times six sides, for a total of… 150 square inches. That's only one-fifth the paper we'd need to wrap each die individually! An animation shows two images. On image shows the dice spread out. The calculation for the surface area of all these dice is "six square inches times 125 equals 750 square inches." The second image shows the dice arranged as a cube. The calculation for the surface area of this arrangement is "25 square inches times six equals 150 square inches." MOBY: Beep! NAT: Yup, a small sheet will do the job! An animation shows Moby wrapping the dice. Again, the wrapped gift looks awful. NAT: We're on a roll—what's next? Two customers approach the wrapping station carrying a live anteater. They hoist the anteater onto the counter. NAT: Wow, they really do have everything here. The anteater starts licking the roll of tape with its long tongue. Then it licks Moby's face. MOBY: Beep! Moby holds up an old-timey watch on a chain. He gestures to the time. NAT: Ooh, look at the time, our shift is just about over. Sorry, Peacock Leader, that's a wrap! Moby sidles away from the booth. He extends an arm and pulls Nat along with him. The disgruntled customers are left alone at the booth. Category:BrainPOP Transcripts Category:BrainPOP Math Transcripts